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According to this definition, and numerous other ones, Karrar is an Arabic word that means one who fights and attacks with power in battle. Also, Karrar is the opposite word of Farrar which is one who escapes cowardly from battle.

This comment summarizes the part of the story of Khaybar where this title was first given to Imam Ali (a); he was named Karrar by the Holy Prophet (s) during the battle of Khaybar.

After a couple of Sahaba bravely fought to open the gate of castle of Khaybar but were not successful. The Holy Prophet (s) said: "Tommorow I will give the flag to a man who Allah loves and his Prophet loves him, and he loves Allah and he loves his Prophet, Karrar and not Farrar." That night the Prophet (s) choose Imam Ali (a), he cured him from his eye sickness that was preventing him from entering the battle. The next day Imam Ali (a) rode into battle and brought down the gate of the castle of Khaybar leading the Muslims into victory. This summary does not provide full justice to the story so it is better read the history on your own; I have provided some places to start, below.

Sources and Further Reading:

The facebook comment provided a good summary. To support its credibility here is some sources you can take a look at. Battle of Khaybar wikipedia page which has a thorough details about the battle make sure to check Fall of al-Qamos fort section. Also this is recorded in Sahih Bukhari 5:51:7 in hadith narrated by Sahl bin Sad. There is also an article on al-islam.org on the Conquest of Khaybar which has a collection of Muslim and non-Muslim writings about the event.